No Customer Left Behind: A Distribution-Free Bayesian Approach to Accounting for Missing Xs in Marketing Models

In marketing applications, it is common that some key covariates in a regression model, such as marketing mix variables or consumer profiles, are subject to missingness. The convenient method that excludes the consumers with missingness in any covariate can result in a substantial loss of efficiency and may lead to strong selection bias in the estimation of consumer preferences and sensitivities. To solve these problems, we propose a new Bayesian distribution-free approach, which can ensure that no customer is left behind in the analysis as a result of missing covariates. In this way, all customers are being considered in devising managerial policies. The proposed approach allows for flexible modeling of a joint distribution of multidimensional interrelated covariates that can contain both continuous and discrete variables. At the same time, it minimizes the impact of distributional assumptions involved in covariate modeling because the method does not require researchers to specify parametric distributions for covariates and can automatically generate suitable distributions for missing covariates. We have developed an efficient Markov chain Monte Carlo algorithm for inference. Besides robustness and flexibility, the proposed approach reduces modeling and computational efforts associated with missing covariates and therefore makes the missing covariate problems easier to handle. We evaluate the performance of the proposed method using extensive simulation studies. We then illustrate the method in two real data examples in which missing covariates occur: a mixed multinomial logit discrete-choice model in a ketchup data set and a hierarchical probit purchase incidence model in a retail store data set. These analyses demonstrate that the proposed method overcomes several important limitations of existing approaches for solving missing covariate problems and offers opportunities to make better managerial decisions with the current available marketing databases. Although our applications focus on consumer-level data, the proposed method is general and can be applied to other marketing applications where other types of marketing players are the units of analysis.